In order to linearize a rational matrix \(R(\lambda )\in \mathbb {C}(\lambda )^{m \times n}\) , we consider in this paper linear polynomial system matrices of \(R(\lambda )\)  [1]}. This means that we consider block partitioned pencils
\(L(\lambda ):= \left[\begin{array}{ccc} \lambda A_1 -A_0 & - \lambda B_1 + B_0 \\ \lambda C_1 -C_0 & \lambda D_1-D_0 \end{array}\right] =: \begin{bmatrix}A(\lambda ) & -B(\lambda )\\C(\lambda ) & D(\lambda )\end{bmatrix} \in \mathbb {C}[\lambda ]^{(p+m) \times (p+n)},\) 
